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Theorem etransclem27 39785
Description: The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem27.s (𝜑𝑆 ∈ {ℝ, ℂ})
etransclem27.x (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem27.p (𝜑𝑃 ∈ ℕ)
etransclem27.h 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem27.cfi (𝜑𝐶 ∈ Fin)
etransclem27.cf (𝜑𝐶:dom 𝐶⟶(ℕ0𝑚 (0...𝑀)))
etransclem27.g 𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))
etransclem27.jx (𝜑𝐽𝑋)
etransclem27.jz (𝜑𝐽 ∈ ℤ)
Assertion
Ref Expression
etransclem27 (𝜑 → (𝐺𝐽) ∈ ℤ)
Distinct variable groups:   𝐶,𝑗,𝑙,𝑥   𝑥,𝐻   𝑗,𝐽,𝑙,𝑥   𝑗,𝑀,𝑥   𝑃,𝑗,𝑥   𝑥,𝑆   𝑗,𝑋,𝑥   𝜑,𝑗,𝑙,𝑥
Allowed substitution hints:   𝑃(𝑙)   𝑆(𝑗,𝑙)   𝐺(𝑥,𝑗,𝑙)   𝐻(𝑗,𝑙)   𝑀(𝑙)   𝑋(𝑙)

Proof of Theorem etransclem27
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem27.g . . . 4 𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥)))
3 fveq2 6148 . . . . . 6 (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
43prodeq2ad 39228 . . . . 5 (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
54sumeq2ad 39201 . . . 4 (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
65adantl 482 . . 3 ((𝜑𝑥 = 𝐽) → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
7 etransclem27.jx . . 3 (𝜑𝐽𝑋)
8 etransclem27.cfi . . . . 5 (𝜑𝐶 ∈ Fin)
9 dmfi 8188 . . . . 5 (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
108, 9syl 17 . . . 4 (𝜑 → dom 𝐶 ∈ Fin)
11 fzfid 12712 . . . . 5 ((𝜑𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
12 etransclem27.s . . . . . . . 8 (𝜑𝑆 ∈ {ℝ, ℂ})
1312ad2antrr 761 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
14 etransclem27.x . . . . . . . 8 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
1514ad2antrr 761 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
16 etransclem27.p . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
1716ad2antrr 761 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
18 etransclem27.h . . . . . . . 8 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
19 etransclem5 39763 . . . . . . . 8 (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
2018, 19eqtri 2643 . . . . . . 7 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦𝑋 ↦ ((𝑦𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
21 simpr 477 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
22 etransclem27.cf . . . . . . . . . 10 (𝜑𝐶:dom 𝐶⟶(ℕ0𝑚 (0...𝑀)))
2322ffvelrnda 6315 . . . . . . . . 9 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)))
24 elmapi 7823 . . . . . . . . 9 ((𝐶𝑙) ∈ (ℕ0𝑚 (0...𝑀)) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2523, 24syl 17 . . . . . . . 8 ((𝜑𝑙 ∈ dom 𝐶) → (𝐶𝑙):(0...𝑀)⟶ℕ0)
2625ffvelrnda 6315 . . . . . . 7 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℕ0)
2713, 15, 17, 20, 21, 26etransclem20 39778 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗)):𝑋⟶ℂ)
287ad2antrr 761 . . . . . 6 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽𝑋)
2927, 28ffvelrnd 6316 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3011, 29fprodcl 14607 . . . 4 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
3110, 30fsumcl 14397 . . 3 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℂ)
322, 6, 7, 31fvmptd 6245 . 2 (𝜑 → (𝐺𝐽) = Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽))
3313, 15, 17, 20, 21, 26, 28etransclem21 39779 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))))
34 iftrue 4064 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) = 0)
35 0zd 11333 . . . . . . . 8 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → 0 ∈ ℤ)
3634, 35eqeltrd 2698 . . . . . . 7 (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
3736adantl 482 . . . . . 6 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
38 0zd 11333 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ∈ ℤ)
39 nnm1nn0 11278 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
4016, 39syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 − 1) ∈ ℕ0)
4116nnnn0d 11295 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ ℕ0)
4240, 41ifcld 4103 . . . . . . . . . . . . . . 15 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
4342nn0zd 11424 . . . . . . . . . . . . . 14 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4443ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
4526nn0zd 11424 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4645adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℤ)
4744, 46zsubcld 11431 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ)
4838, 44, 473jca 1240 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ))
4946zred 11426 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5044zred 11426 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
51 simpr 477 . . . . . . . . . . . . . 14 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗))
5249, 50, 51nltled 10131 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
5350, 49subge0d 10561 . . . . . . . . . . . . 13 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ↔ ((𝐶𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)))
5452, 53mpbird 247 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))
55 0red 9985 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
5626nn0red 11296 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶𝑙)‘𝑗) ∈ ℝ)
5742nn0red 11296 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5857ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
5926nn0ge0d 11298 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶𝑙)‘𝑗))
6055, 56, 58, 59lesub2dd 10588 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
6158recnd 10012 . . . . . . . . . . . . . . 15 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
6261subid1d 10325 . . . . . . . . . . . . . 14 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
6360, 62breqtrd 4639 . . . . . . . . . . . . 13 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6463adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
6548, 54, 64jca32 557 . . . . . . . . . . 11 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
66 elfz2 12275 . . . . . . . . . . 11 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) ↔ ((0 ∈ ℤ ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ) ∧ (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))))
6765, 66sylibr 224 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)))
68 permnn 13053 . . . . . . . . . 10 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℕ)
6967, 68syl 17 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℕ)
7069nnzd 11425 . . . . . . . 8 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℤ)
71 etransclem27.jz . . . . . . . . . . 11 (𝜑𝐽 ∈ ℤ)
7271ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
73 elfzelz 12284 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
7473ad2antlr 762 . . . . . . . . . 10 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
7572, 74zsubcld 11431 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (𝐽𝑗) ∈ ℤ)
76 elnn0z 11334 . . . . . . . . . 10 ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0 ↔ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℤ ∧ 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))
7747, 54, 76sylanbrc 697 . . . . . . . . 9 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0)
78 zexpcl 12815 . . . . . . . . 9 (((𝐽𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
7975, 77, 78syl2anc 692 . . . . . . . 8 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))) ∈ ℤ)
8070, 79zmulcld 11432 . . . . . . 7 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) ∈ ℤ)
8138, 80ifcld 4103 . . . . . 6 ((((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
8237, 81pm2.61dan 831 . . . . 5 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗)))) · ((𝐽𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶𝑙)‘𝑗))))) ∈ ℤ)
8333, 82eqeltrd 2698 . . . 4 (((𝜑𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8411, 83fprodzcl 14609 . . 3 ((𝜑𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8510, 84fsumzcl 14399 . 2 (𝜑 → Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝐽) ∈ ℤ)
8632, 85eqeltrd 2698 1 (𝜑 → (𝐺𝐽) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ifcif 4058  {cpr 4150   class class class wbr 4613  cmpt 4673  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  Fincfn 7899  cc 9878  cr 9879  0cc0 9880  1c1 9881   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  cn 10964  0cn0 11236  cz 11321  ...cfz 12268  cexp 12800  !cfa 13000  Σcsu 14350  cprod 14560  t crest 16002  TopOpenctopn 16003  fldccnfld 19665   D𝑛 cdvn 23534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-icc 12124  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-prod 14561  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537  df-dvn 23538
This theorem is referenced by: (None)
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